E500: “Is the K4^3-free Turán density five ninths?”

Open

Statement	User	Model	Harness	Time
2)V2The 3-graph consisting of the unique triple on three vertices is 3-uniform and K4^3-free.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Does the maximum density of a 3-uniform hypergraph with no complete 3-uniform hypergraph on four vertices ten…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The 3-graph consisting of the unique triple on three vertices is 3-uniform and K4^3-free.

kernel-checked, filed Tue Aug 25 2026 06:28:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The smallest nonempty 3-uniform hypergraph under the root's exact edge and forbidden-four-set encoding.

1) V1 Does the maximum density of a 3-uniform hypergraph with no complete 3-uniform hypergraph on four vertices tend to 5/9?

open, filed Tue Aug 25 2026 06:26:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The finite-set encoding matches ex_3(n,K_4^3) directly. Empty and one-edge 3-graphs kernel-check uniformity/K4-freeness boundaries; an independent encoding is definitionally equal; nine content-free bridges fail. Whole routes checked Turán's lower construction, exact flag-algebra certificates, link averaging, stability/symmetrization, and definition degeneracies.

Scope. Turán's exact conjecture in Erdős problem 500. Edges are distinct 3-subsets of Fin n, K4^3-freeness requires a missing triple inside every 4-subset, the finite maximum is normalized by choose(n,3), and the asserted real limit is 5/9.

E500:Erdős Problem #500 “Is the K4^3-free Turán density five ninths?”

E500: “Is the K4^3-free Turán density five ninths?”